General Details Acknowledgements

Prof Waters Mathematical Physiology Notes. 1 General Details Acknowledgements The following notes are based on material generously passed to me by Professors Gaffney, Chapman and Fowler and Dr Hinch, which I gratefully acknowledge. Please pass any errors or comments to waters@maths.ox.ac.uk Website http://www.maths.ox.ac.uk/courses/course/26323 Detailed lecture notes by Hinch, Chapman & Fowler may be found at this site. Research opportunities: http://www.maths.ox.ac.uk/research Background Reading Mathematical Biology and Ecology Part B course J. Keener and J. Sneyd, Mathematical Physiology (Springer-Verlag, 1998), First Edition, or Second edition Vol I: Chs. 2 7. Vol II: Chs. 11, 13, 14. (Springer-Verlag, 2009) Subsidiary mathematical texts are: J. D. Murray, Mathematical Biology (Springer-Verlag, 2nd ed., 1993). [Third edition, Vols I and II, (Springer-Verlag, 2003).] L. Glass and M. C. Mackey, From Clocks to Chaos (Princeton University Press, 1988). P. Grindrod, Patterns and Waves (OUP, 1991). A general physiology text is: R. M. Berne and M. N. Levy, Principles of Physiology (2nd ed., Mosby, St. Louis, 1996). Overview We will initially focus on systems in cellular physiology and electrophysiology where the spatial variation is not present, or at least, not important. Therefore only the temporal evolution needs to be captured in equations and this typically (but not exclusively) leads to ordinary differential equations. We will proceed to consider systems where there is explicit spatial variation, leading to partial differential equation that additionally incorporate spatial effects; our focus will be calcium dynamics and electrophysiology. We will then consider the role of delayed feedback in biological dynamical systems, such as respiration and the hematopoietic system (i.e. the production of blood cells), exploring the counter-intuitive dynamics that emerges.

Prof Waters Mathematical Physiology Notes. 2 1 Transmembrane Ion Transport 1.1 Membrane transport The cell membrane is a phospholipid bilayer separating the cell interior (the cytoplasm) and from the extracellular environment. The membrane contains numerous proteins, and is approximately 7.5nm thick. The most important property of the cell membrane is its selective permeability: it allows the passage of some molecules but restricts the passage of others, thereby regulating the passage of materials into and out of the cell. Many substances penetrate the cellular membrane at rates reflected by their diffusive behaviour in a pure phospholipid bilayer. However, certain molecules and ions such as glucose, amino acids and Na + pass through cell membranes much more rapidly, indicating that the membrane proteins selectively facilitate transport. The membrane contains water-filled pores with diameters of about 0.8nm, and protein-lined pores, called channels or gates, which allow the passage of specific molecules. Both the intracellular and extracellular environments comprise (among other things) a dilute aqueous solution of dissolved salts, mainly NaCl and KCl, which dissociate into Na +, K + and Cl ions. The cell membrane acts as a barrier to the free flow of these ions and to the flow of water. The many mechanisms for facilitating transport across the cellular membrane can be divided up into active and passive processes. An active process is one which requires the expenditure of energy, while a passive process is one which results solely from the random motion of molecules, for example, diffusion. Passive mechanisms by which molecules are transported across the cell membrane include osmosis, diffusion, and carrier-mediated mechanisms. Osmosis, i.e. the diffusion of water down its concentration gradient, is the most important mechanism by which water is transported across the cell membrane. Simple diffusion accounts for the passage of small molecules (e.g. Cl ) through pores and or lipid-soluble molecules (such as oxygen and carbon dioxide) through the lipid bilayer. Carrier-mediated diffusion refers to a process by which a molecule hitches a lift by binding to a carrier molecule which is lipid soluble and can move readily through the membrane. Carriermediated transport occurs when a protein which sits in the membrane has an active site which may be exposed either on the exterior or interior side of the membrane depending on the conformational state of the protein. A substrate (e.g. glucose and amino acids) may bind to the protein in one conformation: the protein then undergoes a conformational change, and the substrate unbinds on the other side of the membrane. The concentration differences that exist between the intracellular and extracellular environments are set up and maintained by active processes. One of the most important of these is the Na + -K + pump, which uses the energy stored in ATP molecules to pump Na + out of the cell and extra intra Na + 437 50 K + 20 397 Table 1: Typical intracellular and extracellular ionic concentrations for the squid giant axon. Units are mm=millimolar= 10 3 M. 1M= 1 molar = 1 mole litre 1.

Prof Waters Mathematical Physiology Notes. 3 K + in. There are also a variety of exchange pumps which use the concentration gradient of one ion to pump another ion against its concentration gradient, such as the Na + -Ca 2+ exchanger, which removes Ca 2+ from the cell at the expense of allowing Na + in. Differences in interior and exterior ionic concentrations create a potential difference across the cell which also drives an ionic current down ion-specific membrane channels. Typical intracellular and extracellular ionic concentrations are shown in Table 1. Figure 1: Representation of an exchange pump 1.1.1 Passive transport: Carrier Mediated Transport Carrier-mediated transport occurs when a protein which sits in the membrane has an active site which may be exposed either on the exterior or interior side of the membrane depending on the conformational state of the protein. A substrate may bind to the protein in one conformation, the protein undergoes a conformational change, and the substrate unbinds on the other side of the membrane. We describe here a simple model for carrier mediated transport. We suppose that the carrier has two conformational states, and that in the first state, labelled C i, the substrate binding site is exposed on the cell interior, while in the second state, labelled C e, the substrate binding site is exposed on the cell exterior (see Figure 2). We suppose that substrate molecules outside the cell (concentration S e ) can bind to C e to produce a complex P e, and that the substrate molecules inside the cell (concentration S i ) can bind to C i to produce a complex P i. Furthermore, we assume that P i can conformally change into P e and vice versa, and at the same rate as the conformational changes of C i and C e (so that the binding of the substrate does not affect the conformational changes of the protein). Thus the reaction scheme is S i + C i k + k P i k k P e k k + S e + C e (1)

Prof Waters Mathematical Physiology Notes. 4 Figure 2: A schematic of the phospholipid membrane double layer, with a gating protein in one of two configurations, C e and C i, spanning the membrane, as part of a carrier mediated transport system. C i k k C e, (2) where we have assumed that the binding affinity of S i to C i is the same as that of S e to C e, and that the two conformational states C i and C e are equally likely. To avoid the system simply settling down to a steady state with zero flux, we assume that the substrate is supplied at a rate J to the exterior and taken away at the same rate from the interior, and we wish to determine this flux of substrate through the membrane as a function of the interior and exterior concentrations. This may be achieved using mass action kinetics (see Mathematical Biology and Ecology), and the computation is carried out on Problem sheet 1, question 1. 1.1.2 Active transport: the sodium-potassium pump The carrier-mediated transport described above moves molecules down chemical gradients. Any process that works against chemical or electrical gradients requires the expenditure of energy, and is known as an active transport mechanism. We now give an example of such a process. First we digress and introduce a little background terminology. Metabolism is the process of extracting useful energy from chemical bonds. The common carrier of energy in the cell is the chemical adenosine triphosphate (ATP). ATP is formed by the addition of an inorganic phosphate group (HPO 2 4 ) to adenosine diphosphate (ADP). The process of adding a phosphate group to a molecule is called phosphorylation. Since the three phosphate groups on ATP carry negative charges, considerable energy is required to overcome the natural repulsion of like-charged phosphates as additional groups are added. Thus, the hydrolysis (the cleavage of a bond by water) of ATP to ADP releases large amounts of energy. The Na + -K + pump pumps sodium ions out the cell against a steep electrochemical gradient, while pumping potassium ions in. This pump alone consumes almost a third of the energy re-

Prof Waters Mathematical Physiology Notes. 5 quirement of a typical animal cell. The pump uses the energy stored in ATP which is released when it is dephosphorylated into ADP, through the overall reaction scheme ATP + 3Na + i + 2K + e ADP + P i + 3Na + e + 2K + i, (3) where the subscripts i and e denote the intracellular and extracellular quantities respectively. The individual components of the reaction are thought to be as follows. When the carrier protein (Na + -K + ATPase) is in its dephosphorylated state, three sodium binding sites are exposed to the cell s interior. When all three binding sites are filled, the carrier protein is phosphorylated by the hydrolysis of ATP into ADP. This phosphorylation induces a conformational change, so that the sodium binding sites are exposed to the cell exterior, and their binding affinity is reduced, causing the release of sodium ions. At the same time, two potassium sites are exposed to the cells exterior. When potassium ions have bound to these two sites the carrier protein is dephosphorylated, inducing the reverse change in conformation, exposing the potassium binding sites to the cell interior and reducing the binding affinity so that the potassium is released. If we simplify this process slightly, assuming that there is a single binding site for sodium and potassium, leading to a one-to-one exchanger rather than the three-for-two which actually happens, then the detailed reaction scheme is Na + + C k 1 k 1 ATP k p ADP NaC NaCP k 2 k 2 Na + e + CP, (4) CP + K + e k 3 k 3 KCP k 4 k 4 P + KC, (5) KC k 5 k 5 K + i + C, (6) where the carrier protein is represented by C in its unphosphorylated, unbound state, CP in its phosphorylated unbound state, NaC when bound to sodium and unphosphorylated, NaCP when bound to sodium and phosphorylated, KC when bound to potassium and unphosphorylated, and KCP when bound to potassium and phosphorylated. Using mass action kinetics, assuming that intracellular potassium and extracellular sodium are removed at a constant rate J, leads to the steady-state flow of ions through the pump J = J 0 [Na + i ][K + e ] [Na + i ] + α[k + i ] + β[k + e ], (7)

Prof Waters Mathematical Physiology Notes. 6 Figure 3: Schematic diagram of the build up of charge on the cell membrane where J 0 = C 0 k 3 k 4 k 5 k 3 k 3 [P ] + k 3 k 5 + k 4 k 5, (8) α = k 1k p + k 2 k 1 + k 2 k p )k 3 k 4 k 5 k 1 k 2 k p (k 3 k 3 [P ] + k 3 k 5 + k 4 k 5 ), β = k 1k p + k 2 k 1 + k 2 k p )k 3 k 4 k 5 k 1 k 2 k p (k 3 k 3 [P ] + k 3 k 5 + k 4 k 5 ), (9) where C 0 is the total concentration of carrier molecule, and [Na + e ] denotes the concentration of sodium ions in the extracellular medium, etc. Note that for both carrier-mediated transport and active transport (7) pump fluxes are linear in concentrations at small concentrations, but saturate to a maximum value at large concentrations (a feature in common with enzyme catalysed reactions). 1.1.3 The membrane potential Different ion concentrations in the interior and exterior of a cell induce a potential difference across the membrane, which is therefore influenced by the active and passive transport of ions into and out of the cell. The membrane potential is fundamental to the study of electrophysiology in diverse settings such as nerve signal propagation and the coordinated contraction of the heart. To explore these applications in more detail we need to consider the electrical properties of the cell membrane. Suppose we have two reservoirs containing different concentrations of a positively charged ion X +. We suppose that both reservoirs are electrically neutral to begin with, so that there is an equal concentration of a negatively charged ion Y. Now suppose that the reservoirs are separated by a semi-permeable membrane which is permeable to X + but not to Y. Then the difference in concentration of X + on each side will lead to the flow of X + across the membrane. However, because Y cannot diffuse through the membrane this will lead to a build up of charge on one side. This charge imbalance sets up an electric field, which produces a force on the ions opposing further diffusion of X +. it is important to realise that the actual amount of X + which diffuses through the membrane is small, and the excess charge all accumulates near the interface, so that to a good approximation the solutions on either side remain electrically neutral. The potential difference at

Prof Waters Mathematical Physiology Notes. 7 which equilibrium is established and diffusion and electric-field-generated fluxes balance is known as the Nernst potential. We can derive an expression for it as follows. If c denotes the concentration of an ion S then the flux of ions due to diffusion is J = D c, where D is the diffusion coefficient. To this we must add the flux due to the fact that the ion carries a charge and is in the presence of an electric field, which is given by J = uzc φ, (10) z where u is the mobility of the ion (defined as the velocity under a constant unit electric field), z is the valence of the ion (so that z/ z is either +1 or -1 and gives the sign of the force on the ion; positive ions move down potential gradients, negative ions move up potential gradients), and φ is the electric potential, so that φ is the electric field. Thus the total flux is given by J = D c uzc φ, (11) z which is the Nernst-Planck equation. Now, there is a relationship (determined by Einstein) between the ionic mobility and the diffusion coefficient, which is D = urt z F, (12) where R is the universal gas constant, T is the temperature and F is Faraday s constant. Furthermore, since the membrane is thin we can replace the Nernst-Plank equation (11) by the one-dimensional version ( dc J = D dx + zf c ) dφ, (13) RT dx where x is a coordinate normal to the membrane. Now, at equilibrium the flux J is zero, giving or equivalently dc dx + zf c dφ RT dx = 0, (14) 1 dc c dx + zf dφ RT dx = 0. (15) Assuming that the interior of the membrane is at z = 0 while the exterior is at x = L, we can integrate from 0 to L to give so that [log c] L 0 + zf RT [φ]l 0 = 0, (16) φ i φ e = zf ( ) RT log ce. (17) c i

Prof Waters Mathematical Physiology Notes. 8 We follow the standard convention of defining the potential difference across the cell membrane as V = φ i φ e, then the Nernst potential for S is V S = zf RT log c e c i = zf RT log [S e] [S i ]. (18) Using the values of the intracellular and extracellular concentrations given in Table 1, typical Nernst potentials for the squid giant axon for potassium and sodium are V K = 77mV (millivolt) and V Na = 56mV. Note that when more than one ion is present, and they have different Nernst potentials, the flux of each individual ion will not be zero even when there is no net current across the membrane. For example, when 77mV < V < 56mVthere will be a flux of K + out of the cell and Na + into the cell through ion specific channels. This flux is balanced by the action of the Na + -K + pump. 1.1.4 Ionic currents The flow of ions across the cell membrane due to concentration differences leads to a build up of charge near the cell membrane and a potential difference across the cell membrane. Thus the cello membrane is effectively acting as a capacitor. The voltage (potential difference) across any capacitor is related to the charge stored Q by where C is the capacitance. V = Q C, (19) Capacitance A simple example of capacitor is two conducting plates, separated by an insulator, for example, an air gap. Connecting a battery to the plates, as illustrated, using wires of low resistance leads to charge flowing onto/off the plates. It will equilibrate extremely quickly. Suppose the charge on the plates is given by +Q eqm and Q eqm respectively. The capacitance of the plates, C, is defined to be C = Q eqm > 0, V where C is a constant, independent of V. Thus the higher the capacitance, the better the plates are at storing charge, for a given potential. The unit of C is the Farad, with 1 Farad equal to 1 Colomb per Volt. If I is the ionic current out of the cell (the rate of flow of positive charges outwards) then the stored charge changes according to Thus, assuming the capacitance is constant, I = dq. (20) C dv This equation is the basis for much theoretical electrophysiology. various models arises in the expression used for the ionic current I. + I = 0. (21) The difference between the

Prof Waters Mathematical Physiology Notes. 9 Figure 4: Schematic diagram of channel gating The simplest model to use is to assume a linear dependence of I on V (as in Ohms law). For a single ion S, with Nernst potential V S, this gives an ionic current I S = g S (V V S ), (22) where the constant g S is the ion-specific membrane conductance, since the current must be zero when V = V S. If more than one ion is present the currents from different ions are simply added together to produce the total ionic current I. 1.1.5 Gating It is found experimentally that g S is not constant but depends on both V and time t. One proposed explanation for this is that the channels are not always open, but may be open or closed, and that the transition rates between open and closed states depends on the potential difference V. The membrane conductance may then be written as g S, where g S is the constant conductance that would result if all the channels were open, and n is the proportion of open channels. For a generic ion, let n be the proportion of open ion channels. Denoting the open channels by O and the closed channels by C, the reaction scheme is simply C α(v ) β(v ) O (23) where α(v ), β(v ) represent voltage dependent rates of switching between the closed and open states. Using a law of mass action we obtain or equivalently dn = α(v )(1 n) β(v )n, (24) τ n (V ) dn = n (V ) n, (25) where n (V ) = α/(α + β) is the equilibrium value of n and τ n (V ) = 1/(α + β) is the timescale for approach to this equilibrium. Both n and τ n can be determined experimentally. 1.1.6 Multiple gates The simple model presented in 1.1.5 can be generalised to channels which contain multiple identical subunits, each of which can be in either the open or closed state.

Prof Waters Mathematical Physiology Notes. 10 Suppose we start by assuming that the channel consists of two gates, which may both exist in one of two states, open or closed. The ion channel is open only if both gates are open; the ion channel is closed if any one gate within the ion channel is closed. We denote S i {0, 1, 2} as the proportion of channels with exactly i gates open. We have S 0 + S 1 + S 2 = 1 (26) and the reaction scheme 2α(V ) S 0 β(v ) α(v ) S 1 2β(V ) S 2. (27) The 2s arise because there are two possible states with one gate open and one gate closed. Since each gate is identical we have lumped these two states into one variable S 1. Using mass action kinetics gives ds 0 ds 2 = β(v )(1 S 0 S 2 ) 2α(V )S 0 (28) = α(v )(1 S 0 S 2 ) 2β(V )S 2 (29) where, in general, V is a function of time (and possibly space too). We could also write down an equation for S 1, but this equation is superfluous since S 1 can be determined from (26). Let n denote the proportion of open gates; thus dn Simple substitution shows that (26), (28) and (29) are satisfied by = α(v )(1 n) β(v )n. (30) S 0 = (1 n) 2, S 1 = 2n(1 n), S 2 = n 2. (31) In fact, it is possible to derive a stronger result. Suppose so that S 1 = 2n(1 n) y 0 y 2. It then follows that S 0 = (1 n) 2 + y 0, S 2 = n 2 + y 2, (32) dy 0 = 2αy 0 β(y 0 + y 2 ), dy 2 = α(y 0 + y 2 ) 2βy 2. (33) Figure 5: Schematic diagram of two identical gate units

Prof Waters Mathematical Physiology Notes. 11 This linear system has eigenvalues (α + β), 2(α + β), and so y 0, y 2 decay exponentially to zero. Thus, regardless of the initial condition, the solution will still approach exponentially that given by (30) and (31) (i.e. (30) and (31) defines a stable invariant manifold of the full system (26), (28) and (29). The analysis of a two-gated channel generalised easily to channels containing more gates. In the case of k identical gates the fraction of open channels is n k, where n again satisfies (30). We will find that a model with 4 gates agree with empirical observations of K + channels, and will be used below. 1.1.7 Non-identical gates Often channels are controlled by more than one protein, with each protein controlling a set of identical gates, but with the gates of each protein different and independent. Consider, for example, the case of a channel with two types of gate, m and h say, each of which may be open or closed. To illustrate we will assume that the channel has two m subunits and one h subunit. With S ij the proportion of channels with i {0, 1, 2} of the m-gates open and j {0, 1} of the h-gates open, the reaction scheme is Simple substitution shows that the corresponding law of mass action equations are satisfied by S 00 = (1 m) 2 (1 h), S 10 = 2m(1 m)(1 h), S 20 = m 2 (1 h), (34) S 01 = (1 m) 2 h, S 11 = 2m(1 m)h, S 21 = m 2 h, (35) so that the proportion of open channels is m 2 h, provided dm/ = α(v )(1 m) β(v )m (36) dh/ = γ(v )(1 h) δ(v )h. (37) As before, such solutions form a stable invariant manifold. We now have the framework in place to enable us to start to model nerve signal propagation.

Prof Waters Mathematical Physiology Notes. 12 Figure 6: A schematic of an unmyelinated neuron, i.e. nerve cell. 1.2 Hodgkin-Huxley model The nervous system is a communication system formed by nerve cells called neurons. Information is propagated along long cylindrical segments called axons by electrochemical signals. An important property of neurons is excitability. If a small current is applied to the cell for a short time, then the membrane potential changes slightly, but returns directly to its equilibrium potential (the resting potential) once the applied current is removed. However, is a sufficiently large current is applied for a short time the membrane potential undergoes a large excursion (called an action potential) before returning to its resting value. It is by the propagation of such action potentials along the axons of neurons that signals are transmitted. The most important landmark in the study of the generation and propagation of signals is the work by Hodgkin and Huxley, who developed the first quantitative model of the propagation of an electrical signal along a squid giant axon (where giant here refers to the size of the axon, not the squid!). This work is remarkable, not only due to its influence on electophysiology, but also for its stimulation of a new field in applied mathematics, the study of excitable systems, that has resulted in a vast amount of research. In the next Chapter we will consider the propagation of action potentials along the axon. In building up to that model we first consider the Hodgkin-Huxley model of the excitability of an axon without the extra complication of spatial variation in the membrane potential. Such a system can be realised experimentally by inserting a thin electrode along the centre of the axon. The interior of the axon will quickly equilibrate, and there will be no spatial variation in the membrane potential, or any current, along the inside of the axon. This is known as the space clamp technique. If we suppose we are applying an external inward current I ext to the cell (for example due to the experimental injection of electrolytes into the axon), then the total outward current is I i I ext, where I i is the outward ionic current as before. Then equation (21) gives C dv + I i I ext = 0, (38) where C is the membrane capacitance. In the squid giant axon, as in many neurons, the most important ionic currents are due to the movement of sodium and potassium ions. Experimentally it is found that the potassium conductance may be modelled by a gated channel

Prof Waters Mathematical Physiology Notes. 13 of the form described in 1.1.5 with an exponent of 4, so that the conductance is g K n 4 where τ n (V ) dn = n (V ) n. (39) While this seems to imply that the potassium channel is controlled by a protein with four identical gates, the exponent is actually determined as a reasonable fit to measured ionic currents, and not from any detailed physiological knowledge of the potassium channel itself. The equilibrium value n is found to be an increasing function of V. Thus at elevated potentials, n is increased, thereby turning on, or activating, the potassium current. The Nernst potential for potassium is below the resting potential, so that the potassium current is an outward current at potentials greater than the resting potential. The function n(t) is called the potassium activation. For the sodium conductance, experimental data suggests that there are two processes at work, one which turns on the sodium current and one which turns it off. The model of Hodgkin-Huxley assumes a channel controlled by proteins, with a conductance of the form g Na m 3 h where τ m (V ) dm = m (V ) m, τ h (V ) dh = h (V ) h. (40) Again, while this seems to imply a channel with three m gates and one h gate, the exponents are determined as a reasonable fit to measured ionic currents, and not from a detailed knowledge of the sodium channel itself. At the resting potential, m is small and h is close to one. For elevated potentials h decreases and m increases. Thus m is called the sodium activation, and h is called the sodium inactivation. The Nernst potential for sodium is above the resting potential, so that the sodium current is an inward current at potentials greater than rest. In (38), I i is the ionic current into the exterior, which is made up from sodium, potassium and leakage contributions. The leakage current across the membrane is the current which is not due to ion channels, and thus is not ideally named. It is linear but is not zero at V = 0 (due to ion pumps for example). Thus where g L and V L are constants. The Hodgkin-Huxley model is I L = g L (V V L ), (41) I i = I Na + I K + I L = g Na m 3 h(v V Na ) + g K n 4 (V V K ) + g L (V V L ), (42) Aside V Na, V K, V L will vary with temperature as this will alter the value of the transmembrane potential where the ion currents are zero (look up the Nernst equation and the Nernst potential); below we only consider a constant temperature. Understanding the effect of varying temperature is beyond the scope of these lectures. Summary In this section, we have the framework for modelling the temporal variations of a space clamped axon plasma membrane potential, which thus allows us to model nerve signal propagation.

Prof Waters Mathematical Physiology Notes. 14 1.2.1 The Huxley Hodgkin model for a space clamped axon The Hodgkin Huxley equations for an axon are given by τ m (V ) dm τ h (V ) dh τ n (V ) dn 0 = C dv + g Nam 3 h(v V Na ) + g K n 4 (V V K ) + g L (V V L ) I ext (43) = m (V ) m, (44) = h (V ) h, (45) = n (V ) n. (46) Note that equation (43) can be rewritten as where C dv = I ext (g Na m 3 h + g K n 4 + g L )(V V eqm ) (47) V eqm = g Nam 3 hv Na + g K n 4 V N + g L V L g Na m 3 h + g K n 4 + g L (48) is the resting potential or equilibrium potential. It is convenient to rewrite the Hodgkin Huxley equations in terms of the deviation of the cell plasma membrane potential from the resting potential ν = V V eqm, where the resting potential V eqm = 70mV, in the following form where τ m (ν) dm τ h (ν) dh τ n (ν) dn 0 = C dν + g Nam 3 h(ν ν Na ) + g K n 4 (ν ν K ) + g L (ν ν L ) I ext (49) = m (ν) m (50) = h (ν) h (51) = n (ν) n (52) ν K = V K V eq = 12mV, ν Na = V Na V eq = 115mV, ν L = V L V eq = 10.6mV, g Na = 0.12Ω 1 cm 3, g K = 0.036Ω 1 cm 3, g L = 3 10 4 Ω 1 cm 3, C = 1µFarad cm 2 with Ω denoting the Ohm, that is the unit of resistance. Note that the system is at equilibrium when ν = 0. We also have introduced m (ν) = α m (ν) α m (ν) + β m (ν) τ m (ν) = 1 α m (ν) + β m (ν) (53)

Prof Waters Mathematical Physiology Notes. 15 where and similarly for h (ν), n (ν), τ h (ν), τ n (ν). Given the properties of the gating functions α m (ν) = α m (V ), β m (ν) = β m (V ), (54) m (ν), h (ν), n (ν), τ m (ν), τ h (ν), τ n (ν), we can see understand the qualitative form of a nerve pulse in the Huxley Hodgkin model for a space clamped axon. The membrane is excitable. The resting potential, which now corresponds to ν = 0 is stable for a sufficiently small perturbation to the system. A finite perturbation causes an excursion in the membrane potential away from the resting potential called an action potential. Suppose we consider a large increase of ν away from the equilibrium (resting) potential of, say, 10mV. From figure 7 we see that τ m is small. Hence, from equation (50) we see that m can be approximated by its pseudo-steady state on the timescale of a nerve pulse and we can take m m (ν). (55) Thus m increases as ν increases, and the conductivity of the membrane with respect to Na + increases. Now since ν ν Na < 0, and positive ions flow move down potential gradients, then sodium ions flow into the cell and the membrane potential ν increases. Hence m continues to increase, and we initially have a positive feedback loop. However, as ν increases, h drops and h approaches h (ν) turning off the Na + current, though on a slower timescale as τ h is larger. As ν increases, n increases and the membrane conductivity with respect to K + increases, though again on a slower timescale. Now, since ν ν K > 0 potassium ions flow out of the cell and we have that ν decreases. Now as ν drops, n becomes smaller. The Na + current remains essentially off as m m (ν), small, for small ν. The K + current is also turned off as n (ν) reduces, but on a slow timescale. Hence the potential, ν drops below zero before resetting to the equilibrium value ν = 0. Figure 7: Graphs of the gating functions m (ν), h (ν), n (ν), τ m (ν), τ h (ν), τ n (ν).

Prof Waters Mathematical Physiology Notes. 16 1.2.2 Quantitative behaviour and model simplification The observation that τ m (ν) is smaller than all other timescales entails that m can be approximated by its pseudo-steady state on the timescale of a nerve pulse and we can take m m (ν), (56) and this can be confirmed numerically. We can also plot the variables n(t), h(t) during the action potential; a core approximation one can deduce from this figure is that n + h 0.85 (57) is a reasonable approximation for the duration of the action potential. Why this occurs can also be seen as follows. Add the equations for n, h to obtain [ ] d τ n (ν) (n + h) + [τ h (ν) τ n (ν)] dh = (n (ν) + h (ν)) (n + h), Note also that n (ν) + h (ν) const = 0.85 for all ν. We thus have the (a posteriori) observation that τ n, (τ h τ n ) are sufficiently small that the term (τ h (ν) τ n (ν)) dh does not sufficiently influence the dynamics to prevent n + h being reasonably approximated by its asymptotic V dependence, n + h. Given these approximations the Huxley Hodgkin equations reduce to the two dimensional system C dν dn = g Na m 3 (ν)(0.85 n)(ν ν Na ) g K n 4 (ν ν K ) g L (ν ν L ) + I ext (58) = α n (ν)(1 n) β n (ν)n, (59) Figure 8: The gating variables during a Huxley Hodgkin equations predicted action potential, via numerical solution.

Prof Waters Mathematical Physiology Notes. 17 with the experimentally determined parameter values g Na = 0.120Ω 1 cm 3, g K = 0.036Ω 1 cm 3, g L = 3 10 4 Ω 1 cm 3, ν Na = 115mV, ν K = 12mV, ν Na = 10.6mV, C = 1µFarad/cm 2, where Ω is one Ohm, the unit of resistance. We now non-dimensionalise with v = ν/ν Na and τ = t/[5ms], as τ n (ν) O(5ms), and we have ɛ dv dτ dn dτ = I ext g(v, n) def = I ext [ m 3 (v)(0.85 n)(v 1) + γ K n 4 (v + v K ) + γ L (v v L ) ] (60) = k(v, n) def = 1 τ n(v) [n (v) n], (61) with ɛ = C g Na [5ms] = 2 10 3, γ K = g K g Na 0.3, γ L = g L g Na 3 10 3, I ext = I ext g Na ν Na, v K = ν K ν Na 0.1, v L = ν L ν Na 0.1, τ n(v) = τ n (ν)/[5ms] O(1). We can now sketch the nullclines, and noting the different timescales of the membrane potential response to the gating variable response (as ɛ 1) we can consider the phase plane trajectories in terms of fast and slow dynamics. Plotting nullclines We set I ext here. The nullclines are curves on which dn/ = 0 and dv/ = 0.

Prof Waters Mathematical Physiology Notes. 18

Prof Waters Mathematical Physiology Notes. 19 Classification of the equilibrium point One can deduce that the stationary point is stable from qualitative features of the phase plane without resorting to the excessive calculation. Perturbing around the stationary point as follows n = n (0) + ˆn, v = ˆv (62) where 0 < ˆn, ˆv 1 and linearising gives ( d ˆv ˆn ) = ( gv /ɛ g n /ɛ k ν k n ) ( ˆv ˆn ) ( def ˆv = M ˆn ) (63) Recall that the stationary point is linearly stable if and only if Tr(M) = g v /ɛ + k n < 0, ɛdet(m) = g v k n + g n k v > 0. (64) Hence we have that the stationary point is linearly stable.

Prof Waters Mathematical Physiology Notes. 20 1.3 Fitzhugh-Nagumo equation 1.3.1 The Fitzhugh Nagumo equations Here the nullclines associated with the functions g(n, v) and k(n, v) are approximated by, respectively, a cubic polynomial and a straight line.

Prof Waters Mathematical Physiology Notes. 21 In particular, if we let w = n n eqm, the Fitzhugh Nagumo equations are ɛ dv dτ = Av(v a)(1 v) w + I ext, dw dτ = w + bv, where 0 < A O(10), a (0, 1) and, for excitable systems, b is sufficiently large to guarantee there is only one equilibrium point. These have a directly analogous phase behaviour to the Huxley-Hodgkin equations, though the quantitative details are different. In the example sheets, you will also consider the possibility of oscillations rather than excitable behaviour in the Fitzhugh-Nagumo equations once I ext 0.